Adapting free-space fast multipole method for layered media Green's function: Algorithm and analysis

We present a numerical algorithm for an accurate and efficient computation of the convolution frequency domain layered media Green's function with given density function. Instead compressing matrix directly as in classical fast multipole, direct solver, H-matrix algorithms, this new considers translated form original so existing blocks from highly optimized free-space multipole method can be easily adapted to An asymptotic analysis is performed on Sommerfeld integrals provide estimate decay rate “multipole” “local” expansions. To avoid oscillatory integrand integral representations when source target are close each other, mathematically equivalent alternative direction introduced. The convergence expansions quadrature rules numerically validated.